A rectangle with side length a and b are in ratio $a : b = (n+1)^2 : n^2$, where n is a positive integer.
Is it possible to cut each such rectangle into two pieces, which can be put together to build a square?
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The answer to your question:
Yes, this is always possible
This solution, shown with $n = 6$, can be applied for all $n$. The rectangle with dimensions $n^2$ by $(n+1)^2$ is divided into $n\times(n+1)$ congruent rectangles with dimensions $n$ by $n+1$ such that there are $n$ rows of height $n$ and $n+1$ columns of width $n+1$. The rectangle is cut into two congruent pieces along a stair-step diagonal. These pieces can be arranged to form a square with edge length $n\times(n+1)$, divided into $n+1$ rows of height $n$ and $n$ columns of width $n+1$.